Imaginary numbers are represented as some real number multiplied by the number "i", which is a representation of the square root of -1. So 5.29i is an imaginary number. There are also complex numbers, made up of a real and imaginary part, like 3.5-22.6i.

The number i pops up in many relations. eix=cosx+isinx for instance.

Their uses are many. One example of their use is in damped oscillations. You might think that damped oscillations are a pretty narrow topic, but many things in nature work that way - sound, AC circuits, light in an absorptive medium etc.

Originally posted by Jack What are imaginary numbers and how and why are they used in physics?

Please could you try and make your answers as simple as possible and bear in mind that I have not even finished my GCSE course in maths yet.

Imaginary numbers are all those numbers whose square is a negative real numbers. All this number can be represented by the product of the square root of -1 (usually written as i or j in engineering literature) and a real number. The sum of a real number (positive square) and of an imaginary number is called a complex number. This are the numbers that are used in physics.

Their use is mostly a very useful mathematical tool (this is a disputed subject since there is also who believes that they are actually the 'natural' numbers to use to describe the physical world). Their introduction allows to compact two parameters into one pretty much like using a 2D vector and vector calculus. There is a large amount of very powerful theorems that allows to simplify difficult problem with real number, passing to the complex ones.
Example of this are all phenomena involving oscillations since their complex description is way more compact than the real one -even though it has some limitations. All description of physical systems that display some kind of planar geometry or traslational simmetry can also benefit from this representation since equations get a simpler form. The use of complex number in physics received quite a boost with the introduction of quantum mechanics where complex numbers are the standard while real ones are somewhat exceptional and appear only in what is measurable.